Talk:Student's t-test: Difference between revisions
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==Calculations== |
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I don't suppose anyone wants to add HOW TO DO a t-test?? |
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== Assumptions == |
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:That seems to be a deficiency of a fairly large number of statistics pages. The trouble seems to be that they're getting written by people who've gotten good grades in statistics courses in which the topics are covered, but whose ability does not exceed what that would imply. Maybe I'll be back.... [[User:Michael Hardy|Michael Hardy]] 22:04, 7 June 2006 (UTC) |
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Maybe I'm missing something, but it seems like the assumptions section is extremely wrong. The underlying distributions do *not* need to be normal. The statistics' (i.e., sample average) distributions need to be normally distributed, and they will be, according to the Central Limit Theorem. [[Special:Contributions/70.35.57.149|70.35.57.149]] ([[User talk:70.35.57.149|talk]]) 19:13, 7 March 2017 (UTC) |
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:My understanding is that you are right, mostly. Only for small samples do we need the sample(s) to follow a normal distribution, when the mean (numerator) and standard error (denominator) won't automatically be normally distributed according to the CLT. And this is the situation where t-tests are most important, because when the samples are large enough for the CLT to apply, they're also large enough for the t-distribution to converge to the Z-distribution. I think this ought to be mentioned (although my authority for this is a statistician friend - I'm still looking for a published statement about it). Then the bit that describes how to test a sample for normality brings a special irony, because a test (like the Shapiro-Wilk or Kolmogorov-Smirnov) for normality is more likely to reject the null hypothesis of normality as the sample size becomes larger, and this is exactly when you don't need to worry so much about normality! [[User:RMGunton|RMGunton]] ([[User talk:RMGunton|talk]]) 15:45, 13 February 2019 (UTC) |
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:If I have time to learn TeX, maybe I'll do it. I know the calculations, it's just a matter of getting Wikipedia to display it properly. [[User:Chris53516|Chris53516]] 16:17, 19 September 2006 (UTC) |
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::Those who don't know TeX can present useful changes here on the talk page in ASCII (plain text), and others can translate them into TeX. I can do basic TeX; you can contact me on my talk page to ask for help. (i.e. I can generally translate equations into TeX; I may not be able to help with more advanced TeX questions.) --[[User:Coppertwig|Coppertwig]] 11:57, 8 February 2007 (UTC) |
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:: The sample mean need not be normally distributed either. Sketch of proof: Efron (1969) (Student's t-Test Under Symmetry Conditions) shows in Section 1 that a proof by Fisher (1925) (Applications of "Student's" Distribution) for the normal case actually only uses the 'sphericity / rotational invariance / orthogonal invariance' of the normal distribution of individual observations for the t-test to control size (Type I error). So, orthogonal invariance of the distribution of X := (X_1, X_2, ..., X_n) is sufficient. This absolutely does not imply that the sample mean is normally distributed, so normality of the sample mean is not necessary. For (counter)example, if n = 3 then it follows from Archimedes' Hat-Box Theorem that a random variable distributed uniformly over the unit sphere (which is clearly orthogonal invariant) has a sample mean that follows a uniform distribution. [[User:NWK2|NWK2]] ([[User talk:NWK2|talk]]) 14:31, 3 June 2021 (UTC) |
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:I uploaded some crappy images of the calculations. I don't have time to mess with TeX, so someone that's a little more TeX-savvy (*snicker*) can do it. [[User:Chris53516|Chris53516]] 16:42, 19 September 2006 (UTC) |
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:I added a tag "dubious" to assumptions section. I agree that the distribution does not to be normal. I further think that variance does not have to follow Chi squared distribution. Even if part of it is true, it sounds very misleading. I included Shapiro-Wilk test in an official document before running the t-test, partly because of this Wikipedia page.  |
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:Should these two assumptions be deleted entirely, or should one or both be substituted with some other statements in order to not be misleading? [[Special:Contributions/38.104.28.226|38.104.28.226]] ([[User talk:38.104.28.226|talk]]) 16:10, 17 October 2022 (UTC) |
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== Worked_examples values are confusing == |
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:[[User:Michael Hardy]] converted two of my crappy graphics to TeX, and I used his conversion to do the last. So there you have it, calculations for the ''t''-test. [[User:Chris53516|Chris53516]] 18:21, 19 September 2006 (UTC) |
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::Great. Now, could someone explicit the formulla? I assume than N is the sample size, s the standard deviation, but what is the df1/dft? ... Ok I found the meaning of df. I find the notation a bit comfusing. it looks a lot like the derivative of a function... is dft thez degrees of freedom of the global population? |
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hey there, just wanted to point out that the values in the [[Student's_t-test#Worked_examples|Worked_examples]] present some speed bumps for folks following along with some tools. in excel/google sheets terms, this is the difference in STDEV() versus STDEVP(). some tools, like numpy.std default to the latter so the values end up differing from examples. i will suggest an edit with values that avoid this that follows for the rest of the example, but wanted to flag this first. |
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::What do you mean: "could someone ''explicit'' the formulla (sic)" (emphasis added)? N is the sample size of group 1 or group 2, depending on which number is there; s is the standard deviation; and df is degress of freedom. There is a degree of freedom for each group and the total. The degrees of freedom for each group is calculated by taking the sample size and subtracting one. The total degrees of freedom is calculated by adding the two groups' degrees of freedom or by subtracting the total sample size by 2. I will change the formula to reflect this and remove the degrees of freedom. [[User:Chris53516|Chris53516]] 13:56, 11 October 2006 (UTC) |
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along these lines, it is somewhat confusing that the difference in means just happens to `0.095`, something that is generally a value used for confidence thresholds. i think any suggestion to fix the first point will take care of this too, but a nice to have to avoid confusion for stats newbie's like me who'd be following this page topic. <!-- Template:Unsigned --><span class="autosigned" style="font-size:85%;">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:StevenLinde|StevenLinde]] ([[User talk:StevenLinde#top|talk]] • [[Special:Contributions/StevenLinde|contribs]]) 18:32, 22 May 2022 (UTC)</span> <!--Autosigned by SineBot--> |
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== Is s the SEM or the SD? == |
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==independent samples== |
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Should 'assumptions' include the idea that we assume all samples are independent? This seems like a major omission. |
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s is used as the SEM when defining the test statistic. |
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==history unclear== |
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"but was forced to use a pen name by his employer who regarded the fact that they were using statistics as a trade secret. In fact, Gosset's identity was unknown not only to fellow statisticians but to his employer - the company insisted on the pseudonym so that it could turn a blind eye to the breach of its rules." What breach? Why didn't the company know? If it didn't know, how is it insisting on a pseudonym? |
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s is used as the SD in the equations related to Slutsky's Theorem [[User:Buffaloandrews|Chris Andrews]] ([[User talk:Buffaloandrews|talk]]) 13:53, 25 January 2024 (UTC) |
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==Welch (or Satterthwaite) approximation?== |
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"As the variance of each group is different, the Welch (or Satterthwaite) approximation to the degrees of freedom is used in the test"... |
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Huh? |
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--[[User:dmb000006|Dan]]|<sup>[[User_talk:Dmb000006|(talk)]]</sup> 15:00, 19 September 2006 (UTC) |
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== Table? == |
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This article doesn't mention the ''t''-table which appears to be necessary to make sense of the ''t'' value. Also, what's the formula used to compute such tables? [[User:BenFrantzDale|—Ben FrantzDale]] 15:07, 12 October 2006 (UTC) |
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:I'm not sure which table you are referring to or what you mean by "make sense of the ''t'' value". Perhaps you mean the table for determining whether ''t'' is statistically significant or not. That would be a [[statistical significance]] matter, not a matter of just the ''t''-test. Besides, that table is pretty big, and for the basic meaning and calculation of ''t'', it isn't necessary. [[User:Chris53516|Chris53516]] 15:24, 12 October 2006 (UTC) |
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:I forgot. The calculation for such equations is calculus, and would be rather cumbersome here. It would belong at the [[statistical significance]] article, anyway. That, and I don't know the calculus behind ''p''. [[User:Chris53516|Chris53516]] 15:26, 12 October 2006 (UTC) |
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:: Duah, [[Student's t-distribution]] has the answer to my question. [[User:BenFrantzDale|—Ben FrantzDale]] 14:55, 13 October 2006 (UTC) |
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:::Glad to be of not-so-much [[Wikipedia:This didn't help, what the heck are you talking about?|help]]. :) [[User:Chris53516|Chris53516]] 15:11, 13 October 2006 (UTC) |
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== Are the calculations right? == |
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The article says: |
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:<math>t = {\overline{X}_1 - \overline{X}_2 \over s_{\overline{X}_1 - \overline{X}_2}} |
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\ \mathrm{where}\ s_{\overline{X}_1 - \overline{X}_2} = \sqrt{{\mathrm({N}_1 - 1)\cdot s_1^2 + \mathrm({N}_2 - 1)\cdot s_2^2 \over \mathrm({N}_1 + {N}_2 - 2)}\left({1 \over N_1} + {1 \over N_2}\right)} |
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</math> |
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But if you ignore the -1 and -2, say for the biased estimator or if there are lots of samples, then s simplifies to |
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:<math>s = \sqrt{ s_1^2 / N_2 + s_2^2 / N_1 } |
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</math> |
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This seems backwards. The external links all divide the standard deviation by its corresponding sample size, which is what I was expecting. So I'd guess there's a typo and the article should have: |
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:<math>t = {\overline{X}_1 - \overline{X}_2 \over s_{\overline{X}_1 - \overline{X}_2}} |
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\ \mathrm{where}\ s_{\overline{X}_1 - \overline{X}_2} = \sqrt{{\mathrm({N}_2 - 1)\cdot s_1^2 + \mathrm({N}_1 - 1)\cdot s_2^2 \over \mathrm({N}_1 + {N}_2 - 2)}\left({1 \over N_1} + {1 \over N_2}\right)} |
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</math> |
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Can anyone confirm this? |
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[[User:Bleachpuppy|Bleachpuppy]] 22:14, 17 November 2006 (UTC) |
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:I think it's right as it stands, but I don't have time to check very carefully. When you multiply ''s''<sub>1</sub><sup>2</sup> by ''N''<sub>1</sub> − 1, you just get the sum of squares of deviations from the sample mean in the first sample. Similarly with "2" instead of "1". So the sum in the numerator is the sum of squares due to error for the two samples combined. Then you divide that sum of squares by its number of degrees of freedom, which is ''N''<sub>1</sub> + ''N''<sub>2</sub> − 2. All pretty standard stuff. [[User:Michael Hardy|Michael Hardy]] 23:23, 17 November 2006 (UTC) |
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::... and I think that just about does it; i.e. I've checked carefully. [[User:Michael Hardy|Michael Hardy]] 23:29, 17 November 2006 (UTC) |
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::: Please provide a citation or derivation. I think Bleachpuppy is right that the subscripts have been switched. Suppose <math>N_1 = 30 </math> and <math>N_2 = 10^9</math>, a very large number, and <math>s_1</math> and <math>s_2</math> are of moderate and comparable size (i.e. <math>N_2</math> is a very large number in comparison to any of the other numbers involved). In this case, in effect <math>\overline{X}_2</math> is known almost perfectly, so the formula should reduce to a close approximation of the t-distribution for the case where the sample 1 is being compared to a fixed null-hypothesis mean <math>\mu</math> which in this case is closely estimated by <math>\overline{X}_2</math>. In other words, it should be approximately equal to: |
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::::<math>t = \frac{\overline{X}_1 - \mu}{(\sigma_1/\sqrt{30})}</math> |
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:::But apparently the formula as written does not reduce to this; instead it reduces to approximately: |
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::::<math>t = \frac{\overline{X}_1 - \mu}{(\sigma_2/\sqrt{30})}</math> |
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:::This is claiming that this statistical test depends critically on <math>\sigma_2</math>. But since <math>N_2</math> is a very large number in this example, <math>\sigma_2</math> should be pretty much irrelevant; we know <math>\overline{X}_2</math> with great precision regardless of the value of <math>\sigma_2</math>, as long as <math>\sigma_2</math> is not also a very large number. And the test should depend on the value of <math>\sigma_1</math> but does not. --[[User:Coppertwig|Coppertwig]] 12:45, 19 November 2006 (UTC) |
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::::All I have with me right now is an intro to stat textbook: Jaccard & Becker, 1997. ''Statistics for the behavioral sciences.'' On page 265, it verifies the original formula. I have many more advanced books in my office, but I won't be there until tomorrow. -[[User:Schwnj|'''Nick''']]<font color="#FF9900">[[User talk:Schwnj|<sup style="font-variant: small-caps;">talk</sup>]]</font> 21:02, 19 November 2006 (UTC) |
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::::P.S. none of the external links really have any useful information on them (they especially lack formulas). Everything that I've come across on the web uses the formula as currently listed in the article. -[[User:Schwnj|'''Nick''']]<font color="#FF9900">[[User talk:Schwnj|<sup style="font-variant: small-caps;">talk</sup>]]</font> 21:29, 19 November 2006 (UTC) |
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::::The original formula is also confirmed by Hays (1994) ''Statistics'' p. 326. -[[User:Schwnj|'''Nick''']]<font color="#FF9900">[[User talk:Schwnj|<sup style="font-variant: small-caps;">talk</sup>]]</font> 19:36, 20 November 2006 (UTC) |
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:::::OK! I see what's wrong!! The formula is a correct formula. However, the article does not state to what problem that formula is a solution! I assumed that the variances of the two populations could differ from each other. Apparently that formula is correct if you're looking at a problem where you know the variance of the two distributions is the same, even though you don't know what the value of the variance is. I'll put that into the article. --[[User:Coppertwig|Coppertwig]] 03:33, 21 November 2006 (UTC) |
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I know these calculations are correct; I simply didn't have my textbook to for a citation. Keep in mind that much of the time we strive to have an equal sample size between the groups, which makes the calculation of ''t'' much easier. I will clarify this in the text. – [[User:Chris53516|Chris53516]] <sup>([[User talk:Chris53516|Talk]])</sup> 14:28, 21 November 2006 (UTC) |
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I'm not certain, but it looks like the calculations don't match the graphic formula; n=6 in the problem, but n=8 in the graphic formula. [[User:24.82.209.151|24.82.209.151]] 07:54, 23 January 2007 (UTC) |
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These are wrong, they do not match each other. In the first you need to divide by 2, and in the second, you need to drop the multiplication by (1/n1+1/n2) That makes them match -DC |
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==Extra <sup>2</sup>? == |
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Where the text reads, "Where ''s''<sup>2</sup> is the grand standard deviation..." I can't tell what that two is referring to. It doesn't appear in the formula above or as a reference. [[User:198.60.114.249|198.60.114.249]] 23:29, 14 December 2006 (UTC) |
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:The equation you're looking for can be found at [[standard deviation]]. It was not included in this page because it would be redundant. However, I will add a link to it in the text you read. — [[User:Chris53516|Chris53516]] <sup>([[User talk:Chris53516|Talk]])</sup> 02:38, 15 December 2006 (UTC) |
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:Thanks Chris! [[User:198.60.114.249|198.60.114.249]] 07:23, 15 December 2006 (UTC) |
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== I wanna buy a vowel ...== |
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I may be off my medication or something, but does this make sense to '''anyone'''? : |
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"In fact, Gosset's identity was unknown not only to fellow statisticians but |
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to his employer—the company insisted on the pseudonym so that it could turn |
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a blind eye to the breach of its rules." |
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So Gosset works for Guinness. Gosset uses a pen-name cuz Guiness |
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told him to. But, um ... Guiness doesn't know who he is and doesn't want to know. |
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So they can turn a blind eye. |
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So they told this person - they know not whom - to use the pen-name. |
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I know this was a beer factory and all but ... somebody help me out here. |
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[[User:CeilingCrash|CeilingCrash]] 05:28, 24 January 2007 (UTC) |
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:: I don't know the history, but maybe they promulgated a general regulation: If you publish anything on your research, use a pseudonym and don't tell us about it. [[User:Michael Hardy|Michael Hardy]] 20:13, 3 May 2007 (UTC) |
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::: Maybe it should should read "'''a''' pseudonym" instead of "'''the''' pseudonym". I'm not so sure management did not know his identity, however. My recollection of the history is that management gave him permission to publish this important paper, but only under a pseudonym. Guiness did not allow publications for reasons of secrecy. Can someone research this and clear it up?--[[User:141.149.181.4|141.149.181.4]] 14:45, 5 May 2007 (UTC) |
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Unfortunatly I have no sources at hand, but the story as I heard it is that Guiness had(/has?) regulations about confidentiallity on all processes used in the factory. Since Gosset used his formulas for grain selection, they fell under the regulations, so he couldn't publish. He than published under the pseudonym, probably with non-official knowladge and consent of the company, which officially couldn't recognize the work as to be his, due to the regulations. |
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== a medical editor's clarification == |
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The correct way of expressing this test is "Student ''t''''Italic text'' test". The word "Student" is not possessive; there is no "apostrophe s" on it. The lowercase "t" is always italicized. And there is no hyphen between the "t" and "test". It's simply "Student ''t''''''Italic text'''' test" |
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I'm a medical editor, and this is according the the American Medical Association ''Manual of Style,''''Italic text'' 9th edition. Sorry I don't really know how to change it - I'm more a word person than a technology person. But I just wanted to correct this. Thank you! -- [[User:Carlct1|Carlct1]] 16:40, 7 February 2007 (UTC) |
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:You need to close those comma edits. When you want bold text, close it off like this:<code><nowiki>'''bold'''</nowiki></code>, and it will appear like this: '''bold'''. Please edit your comment above so it makes more sense using this information. — [[User:Chris53516|Chris53516]] <sup>([[User talk:Chris53516|Talk]])</sup> 17:00, 7 February 2007 (UTC) |
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:I'm not sure you are correct about the possessive use. As the article notes, "Student" was Gosset's pen name, which would require a possessive ''s'' after the name; otherwise, what does the ''s'' mean? The italic on ''t'' is left off of the article name because it can't be used in the heading. There are other problems like this all over Wikipedia, and it's a technical limitation. By the way, I see both use of "t-test" and "t test" on the web, and I'm not sure that either are correct. — [[User:Chris53516|Chris53516]] <sup>([[User talk:Chris53516|Talk]])</sup> 17:05, 7 February 2007 (UTC) |
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== Recent edit causing page not to display properly -- needs to be fixed == |
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Re this edit: |
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10:47, 8 February 2007 by 58.69.201.190 |
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I see useful changes here; but it's not displaying properly, and also I suggest continuing to provide the equation for the unbiased estimate in addition to the link to the definition of it. I.e. I suggest combining parts of the previous version with this edit. I don't have time to fix it at the moment. --[[User:Coppertwig|Coppertwig]] 11:53, 8 February 2007 (UTC) |
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Looking at it again, I'm not sure any useful material was added by that edit, (I had been confused looking at the diff display), so I've simply reverted it. --[[User:Coppertwig|Coppertwig]] 13:02, 8 February 2007 (UTC) |
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== Equal sample sizes misses a factor sqrt(n) == |
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The formula with equal sample size should be a special case of the formula with unequal sample size. |
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However, looking at the formula for the t-test with unequal sample size: |
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:<math>t = {\overline{X}_1 - \overline{X}_2 \over s_{\overline{X}_1 - \overline{X}_2}} |
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\ \mathrm{where}\ s_{\overline{X}_1 - \overline{X}_2} = \sqrt{{({N}_1 - 1) s_1^2 + ({N}_2 - 1) s_2^2 \over {N}_1 + {N}_2 - 2}\left({1 \over N_1} + {1 \over N_2}\right)} |
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</math> |
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and setting n=N_1=N_2 yields |
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:<math>s_{\overline{X}_1 - \overline{X}_2} = \sqrt{s_{\overline{X}_1}^2 + s_{\overline{X}_2}^2} / \sqrt{n}</math>. |
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The factor of sqrt(n) should be correct in the limit of large n. |
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However, there might be a problem since one sets N_1 = N_2 which reduces the degree of freedom by one. Does anyone knows the correct answer? |
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[[User:Oliver.duerr|Oliver.duerr]] 09:13, 20 February 2007 (UTC) |
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:: I don't have the answer, but I agree both formulas don't match [[User:128.186.38.50|128.186.38.50]] 15:37, 10 May 2007 (UTC) |
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== Explaining a revert == |
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I just reverted "t tests" from singular back to plural. The reason is that it's introducing a list of different tests, so there is more than one kind of t test. I mean to put this in the edit summary but accidentally clicked the wrong button. --[[User:Coppertwig|Coppertwig]] 19:55, 25 February 2007 (UTC) |
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== Copied from another source? == |
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Why is this line in the text? |
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'''[The author erroneously calculates the sample standard deviation by dividing N. Instead, we should divide n-1, so the correct value is 0.0497].''' |
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To me, this suggests that portions of the article were copied from another, uncited source. If true, this is copyright infringement and needs to be fixed right away. |
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:I can't find any internet-based source for the text in that part of the article. I think the line might be directed toward the author of the Wikipedia article, as it seems to point out an error. I removed it, and will look into the error. -[[User:Schwnj|'''Nick''']]<font color="#FF9900">[[User talk:Schwnj|<sup style="font-variant: small-caps;">talk</sup>]]</font> 00:38, 19 March 2007 (UTC) |
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By the way; the line should be read carefully. It is correct. As this is an estimate based on an estimate, it should have been divided by n-1, so the correct value is 0.0497. Can someone please change this? |
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== Testing Normality == |
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'''The call''': I think it would be appropriate to change the wording "normal distribution of data, tested by ..." as these tests for normality are only good for establishing that the data is not drawn from a normal distribution. |
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'''The backgroud''': Tests for normality (e.g. Shapiro-Wilk test) test the null hypothesis that the data is normally distributed against the alternative hypothesis that it is not normally distributed. Normality cannot be refuted if the null is not rejected, a statement that can only be statistically evaluated by looking at the power of the test. |
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'''The evidence''': Spiegelhalter (Biometrika, 1980, Table 2) shows that the power of the Shapiro-Wilk test can be very low. There are non normal distributions such that with 50 observations this test only correctly rejects the null hypothesis that the data is not normally distributed 8% (!) of the time. |
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'''At least two possible solutions''': (1) Drop the statement that the assumption of normality can be tested. (2) Indicate that one can test if the data is not normally distributed, pointing out that no rejection of normality does not mean that the data is normally distributed due to the low power of these tests. |
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[[User:Schlag|Schlag]] 11:55, 27 June 2007 (UTC) |
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:: If you perform any of these tests before doing a t-test the p-value under the null hypothesis will no longer be uniformly distributed. This entire section is bad statistical advice (although commonly done in practice) [[User:Hadleywickham|Hadleywickham]] 07:27, 9 July 2007 (UTC) |
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Assumptions
[edit]Maybe I'm missing something, but it seems like the assumptions section is extremely wrong. The underlying distributions do *not* need to be normal. The statistics' (i.e., sample average) distributions need to be normally distributed, and they will be, according to the Central Limit Theorem. 70.35.57.149 (talk) 19:13, 7 March 2017 (UTC)
- My understanding is that you are right, mostly. Only for small samples do we need the sample(s) to follow a normal distribution, when the mean (numerator) and standard error (denominator) won't automatically be normally distributed according to the CLT. And this is the situation where t-tests are most important, because when the samples are large enough for the CLT to apply, they're also large enough for the t-distribution to converge to the Z-distribution. I think this ought to be mentioned (although my authority for this is a statistician friend - I'm still looking for a published statement about it). Then the bit that describes how to test a sample for normality brings a special irony, because a test (like the Shapiro-Wilk or Kolmogorov-Smirnov) for normality is more likely to reject the null hypothesis of normality as the sample size becomes larger, and this is exactly when you don't need to worry so much about normality! RMGunton (talk) 15:45, 13 February 2019 (UTC)
- The sample mean need not be normally distributed either. Sketch of proof: Efron (1969) (Student's t-Test Under Symmetry Conditions) shows in Section 1 that a proof by Fisher (1925) (Applications of "Student's" Distribution) for the normal case actually only uses the 'sphericity / rotational invariance / orthogonal invariance' of the normal distribution of individual observations for the t-test to control size (Type I error). So, orthogonal invariance of the distribution of X := (X_1, X_2, ..., X_n) is sufficient. This absolutely does not imply that the sample mean is normally distributed, so normality of the sample mean is not necessary. For (counter)example, if n = 3 then it follows from Archimedes' Hat-Box Theorem that a random variable distributed uniformly over the unit sphere (which is clearly orthogonal invariant) has a sample mean that follows a uniform distribution. NWK2 (talk) 14:31, 3 June 2021 (UTC)
- I added a tag "dubious" to assumptions section. I agree that the distribution does not to be normal. I further think that variance does not have to follow Chi squared distribution. Even if part of it is true, it sounds very misleading. I included Shapiro-Wilk test in an official document before running the t-test, partly because of this Wikipedia page. 
- Should these two assumptions be deleted entirely, or should one or both be substituted with some other statements in order to not be misleading? 38.104.28.226 (talk) 16:10, 17 October 2022 (UTC)
Worked_examples values are confusing
[edit]hey there, just wanted to point out that the values in the Worked_examples present some speed bumps for folks following along with some tools. in excel/google sheets terms, this is the difference in STDEV() versus STDEVP(). some tools, like numpy.std default to the latter so the values end up differing from examples. i will suggest an edit with values that avoid this that follows for the rest of the example, but wanted to flag this first.
along these lines, it is somewhat confusing that the difference in means just happens to `0.095`, something that is generally a value used for confidence thresholds. i think any suggestion to fix the first point will take care of this too, but a nice to have to avoid confusion for stats newbie's like me who'd be following this page topic. — Preceding unsigned comment added by StevenLinde (talk • contribs) 18:32, 22 May 2022 (UTC)
Is s the SEM or the SD?
[edit]s is used as the SEM when defining the test statistic.
s is used as the SD in the equations related to Slutsky's Theorem Chris Andrews (talk) 13:53, 25 January 2024 (UTC)